Search form

Nonequilibrium Statistical Mechanics

Ryugo Kubo

posted on 30 June 2003

reviewed by Joe McCauley

This text provides a readable introduction to Markov processes,
including both Langevin and Fokker-Planck equations, from the
standpoint of typical physical examples. So why review an ancient
book on statistical physics in an age when we're already into
complexity? After all, every econophysicist already knows the
contents of this book. Right? Well, we as physicists are sometimes
good at criticizing the economists for mistakes about equilbrium that
they propagate in their literature, but we generally ignore the same
misconceptions in our own statistical physics literature, and this
book apparently treats stochastic processes from a standpoint that
assumes that stochastic processes are always near equilibrium. Where
does the mistake creep into this otherwise fine text?

There are two mistakes on pages 65-68. The discussion is based on the
stochatic differential equation (sde)

dx=-R(x)dt+D(x,t)^1/2dB(t)

where B(t) is a Wiener process. First, it is claimed that the random force

D(x,t)^1/2dB

is Gaussian with a white spectrum. In general, though, the random
force cannot be stationary unless D is independent of x. The unstated
assumption of the text seems to be that the random force is always
stationary, so that with R(x)
When the diffusion coefficient depends on x (or more generally on
(x,t)) then there can be no approach to equilibrium for the case of
unbounded x, even with R
finance theory so vividly shows. In the lognormal model, e.g., there
exists an equilibrium solution that is easily calculated in closed
algebraic form, but that solution is not reached by time-dependent
solutions of the Fokker-Planck equation. More generally, even if an
equilibrium solution of the Fokker-Planck equation for a variable
diffusion coefficient (or 'local volatility') D(x) 'exists', it
cannot be reached dynamically when the random force is nonstationary.
Demonstrationing that an equilibrium solution 'exists' is meaningless
are useless if the dynamics can't approach that solution. We know now
that the empirical finance distribution requires an (x,t)-dependent
diffusion coefficient, and that the empirical distribution does not
approach equilibrium as time increases. Finance theorists often
massage their data by 'transforming' to a stationary distribution.
This is a terrribly misleading approximation. In general, stationary
distributions do not exist empirically, especially not where
complexity comes into the picture.

A final point of interest: here's how to test empirically for the
stabilizing effect of Adam Smith's Invisible Hand: check to see if
the economic data are stationary. If not, then the 'Invisible Hand is
unreliable. The problem you'll face is that most economic data are
too poor to test reliably for stationarity, or for anything else, for
that matter (the data are generally too easy to fit). But maybe
Bertrand Roehner can help by pointing us to the better data in
nonfinancial economics.